Question: Let $g(x)=-3\log(x)$. Find $g'(x)$. Choose 1 answer: Choose 1 answer: (Choice A) A $-\dfrac{3}{\log(x)}$ (Choice B) B $-\dfrac{3}{\ln(10)x}$ (Choice C) C $-\dfrac{3}{x\log(x)}$ (Choice D) D $-\dfrac{3}{x\ln(x)}$
Solution: The expression for $g(x)$ includes a logarithmic term. Remember that the derivative of the general logarithmic term $\log_a(x)$ (where $a$ is any positive constant and $a\neq 1$ ) is $\dfrac{1}{\ln(a)\cdot x}$. Put another way, $\dfrac{d}{dx}[\log_a(x)]=\dfrac{1}{\ln(a)\cdot x}$. [Is there an easy way to memorize that?] We can use this to find the derivative of the function as shown below. $\begin{aligned} g'(x)&=\dfrac{d}{dx}[-3\log(x)] \\\\ &=-3\dfrac{d}{dx}[\log(x)] \\\\ &=-3\dfrac{d}{dx}[\log_{10}(x)]&&{\gray{\text{Since }\log(x)=\log_{10}(x)}} \\\\ &=-3\cdot\dfrac{1}{\ln(10)x} \\\\ &=-\dfrac{3}{\ln(10)x} \end{aligned}$ In conclusion, $g'(x)=-\dfrac{3}{\ln(10)x}$.